Evaluate the sum $\sum_{k = 1}^{\infty} (2k + 1)x^{2k}$.

We have to assume $% $, otherwise we can’t use the infinite series approximation.

We can calculate $\sum_{k = 1}^{\infty} x^{2k + 1}$ as follows:

Differentiating both sides of the above equation w.r.t $x$: